p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.12D8, C24.87D4, (C2×C8).53D4, C22.89(C2×D8), C2.16(C8⋊7D4), (C22×C4).160D4, C23.930(C2×D4), C22.4Q16⋊29C2, C4.54(C4.4D4), C2.16(C8.D4), C4.18(C42⋊2C2), C22.122(C4○D8), (C22×C8).115C22, (C23×C4).275C22, C2.8(C22.D8), C23.7Q8.20C2, C22.251(C4⋊D4), (C22×C4).1464C23, C4.110(C22.D4), C2.11(C23.20D4), C2.10(C23.11D4), C22.140(C8.C22), C22.120(C22.D4), (C2×C2.D8)⋊10C2, (C2×C4).1373(C2×D4), (C2×C22⋊C8).28C2, (C2×C4).626(C4○D4), (C2×C4⋊C4).149C22, SmallGroup(128,807)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.12D8
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 288 in 128 conjugacy classes, 48 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C22.4Q16, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×C2.D8, C23.12D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×D8, C4○D8, C8.C22, C23.11D4, C8⋊7D4, C8.D4, C22.D8, C23.20D4, C23.12D8
►On 64 points
Generators in S64
(1 5)(2 22)(3 7)(4 24)(6 18)(8 20)(9 36)(10 63)(11 38)(12 57)(13 40)(14 59)(15 34)(16 61)(17 21)(19 23)(25 29)(26 54)(27 31)(28 56)(30 50)(32 52)(33 41)(35 43)(37 45)(39 47)(42 60)(44 62)(46 64)(48 58)(49 53)(51 55)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 44 17 9)(2 43 18 16)(3 42 19 15)(4 41 20 14)(5 48 21 13)(6 47 22 12)(7 46 23 11)(8 45 24 10)(25 64 49 38)(26 63 50 37)(27 62 51 36)(28 61 52 35)(29 60 53 34)(30 59 54 33)(31 58 55 40)(32 57 56 39)
G:=sub<Sym(64)| (1,5)(2,22)(3,7)(4,24)(6,18)(8,20)(9,36)(10,63)(11,38)(12,57)(13,40)(14,59)(15,34)(16,61)(17,21)(19,23)(25,29)(26,54)(27,31)(28,56)(30,50)(32,52)(33,41)(35,43)(37,45)(39,47)(42,60)(44,62)(46,64)(48,58)(49,53)(51,55), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,44,17,9)(2,43,18,16)(3,42,19,15)(4,41,20,14)(5,48,21,13)(6,47,22,12)(7,46,23,11)(8,45,24,10)(25,64,49,38)(26,63,50,37)(27,62,51,36)(28,61,52,35)(29,60,53,34)(30,59,54,33)(31,58,55,40)(32,57,56,39)>;
G:=Group( (1,5)(2,22)(3,7)(4,24)(6,18)(8,20)(9,36)(10,63)(11,38)(12,57)(13,40)(14,59)(15,34)(16,61)(17,21)(19,23)(25,29)(26,54)(27,31)(28,56)(30,50)(32,52)(33,41)(35,43)(37,45)(39,47)(42,60)(44,62)(46,64)(48,58)(49,53)(51,55), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,44,17,9)(2,43,18,16)(3,42,19,15)(4,41,20,14)(5,48,21,13)(6,47,22,12)(7,46,23,11)(8,45,24,10)(25,64,49,38)(26,63,50,37)(27,62,51,36)(28,61,52,35)(29,60,53,34)(30,59,54,33)(31,58,55,40)(32,57,56,39) );
G=PermutationGroup([[(1,5),(2,22),(3,7),(4,24),(6,18),(8,20),(9,36),(10,63),(11,38),(12,57),(13,40),(14,59),(15,34),(16,61),(17,21),(19,23),(25,29),(26,54),(27,31),(28,56),(30,50),(32,52),(33,41),(35,43),(37,45),(39,47),(42,60),(44,62),(46,64),(48,58),(49,53),(51,55)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,57),(40,58)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,57),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,44,17,9),(2,43,18,16),(3,42,19,15),(4,41,20,14),(5,48,21,13),(6,47,22,12),(7,46,23,11),(8,45,24,10),(25,64,49,38),(26,63,50,37),(27,62,51,36),(28,61,52,35),(29,60,53,34),(30,59,54,33),(31,58,55,40),(32,57,56,39)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D8 | C4○D8 | C8.C22 |
| kernel | C23.12D8 | C22.4Q16 | C23.7Q8 | C2×C22⋊C8 | C2×C2.D8 | C2×C8 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 2 | 1 | 1 | 10 | 4 | 4 | 2 |
Matrix representation of C23.12D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 2 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 11 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[2,0,0,0,0,0,0,9,0,0,0,0,0,0,10,10,0,0,0,0,2,7,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,11,11,0,0,0,0,9,6,0,0,0,0,0,0,4,0,0,0,0,0,0,13] >;
C23.12D8 in GAP, Magma, Sage, TeX
C_2^3._{12}D_8 % in TeX
G:=Group("C2^3.12D8"); // GroupNames label
G:=SmallGroup(128,807);
// by ID
G=gap.SmallGroup(128,807);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,718,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations